Transport in Porous Media 55: 243–246, 2004.© 2004 Kluwer Academic Publishers. Printed in the Netherlands.

243

Letter to the Editor

Modeling Low Intensity Coupled TransportProcesses in Fluid-Saturated Porous Media

WALTER ROSE1,∗ and DEREK ROBINSON2

1P.O. Box 2424, Champaign, IL 61825, U.S.A.2University of Illinois, 273 Altgeld Hall, Urbana, IL 61801, U.S.A. e-mail: robinson@math.uiuc.edu

(Received: 15 October 2002; in final form: 20 June 2003)

Here we start by mentioning in passing the famous Onsager Law, which, accordingto DeGroot and Mazur (1962) and followers, is considered to be the correct way tomodel the transport process linear relationships between diffusive fluxes and theirproperly described conjugate thermodynamic forces.

Our major aim, however, is to focus on (and to mention justifications for) theidea that there are also available useful (albeit possibly less theoretically justified)ad hoc linear relationships which provide practical algorithmic descriptions ofother commonplace macroscopic transport processes in porous solid continuumsystems. These include, for example, processes where coupling effects occur thatappear to uniquely involve directly measured non-diffusive fluxes of fluid particlesdescribing the transport of entities such as mass, momentum and/or energy quanta:this is because of the action of the measured attending driving forces. Examplesof the latter are classical viscous and/or interacting capillary coupling cases thatinclude those previously discussed by Rose (1995 et seq). These typically in-clude multi-phase co- and counter-linear coupling affected processes of nearlysingle-component immiscible and incompressible Newtonian fluids of contrast-ing viscosity, through rigid 2D anisotropic but essentially homogeneous porousmedia.

In particular, we will briefly discuss rationales for employing ad hoc empiri-cally, rather than theoretically, justified algorithms, as ways to easily and sensiblymodel certain classes of coupled irreversible fluid phase transport processes. Theseare important processes which are known to occur commonly in porous sediments.They do not seem, however, to correspond, except perhaps superficially, to thoseother important special-case processes like thermo-diffusion which Onsager (1931)and his followers already have considered with such acuity, by making use of thePrinciple of Microscopic Reversibility.

∗Author for correspondence: e-mail: wdrose@uiuc.edu

244 WALTER ROSE AND DEREK ROBINSON

What follows below is an analysis of a general case linear transport systemwhere the number, n, of the inter-relating conjugate forces and fluxes inter-relationships describing particular non-equilibrium coupled transport processes ofinterest is two or greater.

First we let

e =

e1...

en

represent the energy gradients driving forces in the systems being considered: thesegive rise to the conjugate flux vectors: namely

f =

f1...

fn

.

Then we can state: Ae = f , where A = [aij ] is an n × n matrix having as itselements the material response transport coefficients of proportionality referred toso succinctly in the classical Truesdell and Toupin (1960) article.

Suppose that the n force vectors ei are applied to produce n corresponding fluxvectors f

i. These force and flux vectors (since they both are entities that can be

measured in particular cases of interest) are assumed to be known data. The object,of course, is to compute A from the force and flux data imbedded in the relation-ships: Aei = f

i, which form a system of n2 linear equations for the n2 unknown

transport coefficients aij .Our next step is to write:

ei =

e1i

...

eni

and

fi=

f1i

...

fni

.

Now we denote by A the n2-column vector consisting of the elements of A whenordered by columns. Then the linear system may be written

E∗A = F, where F = [f1, f

2, . . . , f

n].

LETTER TO THE EDITOR 245

Here E∗ is the n2× n2 block matrix having the form:

E∗ =

e11I e21I · en1I

e12I e22I · en2I...

......

...

e1nI e2nI · ennI

with I being the n ×n identity matrix. Note that the entries of A in order are[a11a21 · · · an1 a12a22 · · · an2 · · · a1na2n · · · ann ].

We now limit attention to cases where it is a given fact that the n force vectorsare linearly independent. It follows that the matrix

E = [e1, e2, . . . , en]has linearly independent columns, and this implies that E∗ has linearly independentrows. Therefore by standard matrix algebra (cf. Aitken, 1959) E∗ has an inverse,and indeed we may state that (E∗)−1 = (E−1)∗ since (EG)∗ = G∗E∗.

Accordingly we may therefore conclude that the linear system for the aij hasa unique solution, namely: A = (E∗)−1F . In this connection it may be noted thatthere are fast methods of computing inverses of matrices. Also we point out thatthere are many natural choices for the force vectors ei . For example, we mightchoose eij = 0 when i �= j and eii �= 0. Another possibility would be to chooseeii = 0 and eij = a �= 0 for all i �= j .

The main concluding point of interest, however, is to notice that in this Letter wehave started with an indeterminate transport process equation set, namely: Aei =f

i, where A = [aij ], which in expanded form clearly shows that there are basically

n2 equations for the n2 unknown transport coefficients aij that are involved. Thus,given the data ei, f i

we have shown how to measure the transport coefficients. And

this is achieved by invoking the final equation in this Letter, namely: A = (E∗)−1F ,from which the desired values of aij can be easily extracted.

References

Aitken, A. C.: 1959, Determinants and Matrices, 9th edn, Oliver & Boyd, Edinburgh, London.DeGroot, S. R. and Mazur, P.: 1962, Non-Equilibrium Thermodynamics, North-Holland, Amsterdam.Onsager, L.: 1931, Phys. Rev. 37, 405–426 and 38, 2265–3379.Rose et al., W.: 1995 et seq, comprising eight references as follows: (1) Rose, W.: 1995a, Gener-

alized description of multiphase flow through porous media, Proceedings of the 32nd AnnualTechnical Meeting of the Society of Engineering Sciences, pp. 483–484 (extended abstract); (2)Rose, W.: 1995, Ideas about viscous coupling in anisotropic media, a technical note, Transportin Porous Media 18, 87–93; (3) Letter to the Editor, Transport in Porous Media 22 (1996),359–360; (4) Rose, W.: 1997, An upgraded viscous coupling measurement methodology, atechnical note, Transport in Porous Media 28, 221–231; (5) Rose et al., W.: 1999, Describ-ing coupled transport processes in and through fractured rock systems, Proceedings, Dynamicsof Fluids in Porous Rock Systems, Lawrence Berkley National Laboratory Publication LBNL-42718j, pp. 210–223 (extended abstract); (6) Asadi, M., Ghalambor, A., Rose, W. and Shirazi,

246 WALTER ROSE AND DEREK ROBINSON

M.: 2000, Anisotropic permeability measurement of porous media, a 3-dimensional method,SPE Paper 59396, Proceedings of the Asian Pacific SPE Conference on Management Modeling,25–25 April 2000, Yokohama, Japan; (7) Rose, W.: 2001a, Theory of spontaneous versus inducedcapillary imbibition, a technical note, Transport in Porous Media 44, 591–598; and (8) Rose, W.:2001, Modeling forced versus spontaneous capillary imbibition processes commonly occurringin porous sediments, J. Petroleum Sci. Eng. 30, 144–166.

Truesdell, C. and Toupin, R. A.: 1960, Classical Field Theories, Handbuch der Physik III/1, SpringerVerlag, Berlin, pp. 226–793.